Rooting out an approximation
Q) We are working on square numbers and I have created a number of activities to help pupils understand what a square number is and how to work them out.
Can you suggest a 15-minute activity for the whole class for a lesson-ender? This is the first time the advisory teacher has been to one of my lessons and I feel panicky about being watched.
A) Advisory teachers have seen lots of people teaching different topics.
This variety means they are an excellent source of inspiration. An advisory teacher isn't there to judge you but to make helpful suggestions and answer questions as well as providing you with ideas.
An activity that encourages lots of calculations and challenge is one based on a grid of numbers, and pupils have to find the square root of each number (introduce the term "square root" alongside "to square"). Here is an example of a 3 x 3 grid. This activity would work equally well for pairs of pupils, in primary or secondary school. Have the grid prepared before the lesson (in later lessons ask pupils to choose the numbers). It could be on a large piece of plain paper or even drawn with chalk in the playground, where pupils could play as teams. In this grid you can see that I have included decimal fractions, as this provides a challenge for the more able in the class and stimulates discussion. They may also find it helpful to have a pre-drawn table on paper, giving more time to focus on the activity.
When a pupils has given a correct approximate answer, their initials claim that square, and they get a credit.
If they work in pairs, a lot of understanding will come from their discussions on the choice of number. It is sometimes a good idea to pair a less able pupil with a more able friend.
Pupils will need a calculator each. Demonstrate the activity by choosing one of the numbers as an example. Tell them that the square numbers they have been learning about will be helpful to enable them to make a good guess for the answer.
Suppose you demonstrate with 12. Ask them which square number comes before 12 and which one after 12. Show this on the board.
Enter "3 x 3" before the 12 on the table and "4x4" after the 12. Ask pupils if this helps us to find a number to multiply by itself to make 12. Ask them which number they would like to try. If they try 3.5, enter "3.5 x 3.5" into the table and the answer they get after using their calculator: 12.25.
Ask them if this is the right size, then enter "too big" in the last column.
The process is to guide them in their thoughts, encouraging them to ask themselves questions. Sometimes pupils find it difficult to approach a problem in maths because they don't know what questions to ask - they don't know how to reason. Listening to other people's questions is helpful in learning this process.
They can try anything - let them explore. In my example, the next choice is between 3.4 and 3.5. Draw a number line and ask them what number they would expect to go in the middle. I have found it helpful to write 3.4 as 3.40 and 3.5 as 3.50. This helps to visualise 3.45 more easily. Choosing 3.45 leads us 11.9025 - too small. Highlight that part of the number line.
They might then choose 3.47 (I know the answer, as I cheated and pressed the square root button! Don't be tempted to cheat, you will enjoy the challenge much more if you don't!).
The square of 3.47 is 12.0409. Here, we will accept this as a reasonable approximation to 12 as there is a "0" after the decimal point and the 4 is less than a five, which we normally round up. Pupils won't know about this level of approximation but can use the 4 or smaller rule. Write this on the board to remind them of what target would be acceptable.
lPerfect Times has a game for learning the square numbers called Twirling numbers. www.perfect-times.co.uk
RM's Easiteach (maths) for the number lines www.rm.comprimary